On 14 January 1987, at age 17, Claesson made his debut in the top tier Allsvenskan for the club, scoring a brace in a 13–4 win against Örebro SK.

It was followed up by a consecutive title in 1992, defeating IF Boltic 4–3 in the final, in a season where Claesson became the top scorer in Allsvenskan for the first time in his career.

[1] Claesson was awarded the prize as Swedish Bandy Player of the Year in both 1991 and 1992, Vetlanda BK's gold winning seasons.

Remaining among the top Swedish bandy clubs, Vetlanda BK reached the national final in both 1994 and 1995, but lost to Västerås SK and IF Boltic.

[1] He played his first World Championship for Sweden in 1991, where his country lost 3–4 to the Soviet Union in the final, with Claesson being the top scorer of the tournament.

[6] After his playing career ended, Claesson worked as a pundit for Radiosporten, the sport section of Sveriges Radio.

When his contract expired in 2010, he joined Russian club Kuzbass on a one-year deal, becoming only the second Swedish coach ever to work in the country.